We explore the phase diagram of the strongly correlated Hubbard model with intrinsic spin orbit coupling on the honeycomb lattice. We obtain the low energy effective model describing the spin degree of freedom. We study the resulting model within the Schwinger boson and Schwinger fermion approaches. The Schwinger boson approach gives the boundary between the spin liquid phase and the magnetically ordered phases, Neel order and incommensurate Neel order. We find that increasing the strength of the spin orbit coupling, narrows the width of the spin liquid region. The Schwinger fermion approach sheds further light on the nature of the spin liquid phase. We obtain three different candidates for the spin liquid phase within the mean field approximation which are gapless spin liquid, topological Mott insulator, and the chiral spin liquid phases. We argue that the gauge fluctuations and the instanton effect may suppress the first two spin liquids, while the chiral spin liquid is stable against gauge fluctuations due to its nontrivial topology.

Impurities in superconductors and their induced bound states are important both for engineering novel states such as Majorana zero-energy modes and for probing bulk properties of the superconducting state. The high-temperature cuprates offer a clear advantage in a much larger superconducting order parameter, but the nodal energy spectrum of a pure d-wave superconductor only allows virtual bound states. Fully gapped d-wave superconducting states have, however, been proposed in several cuprate systems thanks to subdominant order parameters producing d + is- or d + id′-wave superconducting states. Here we study both magnetic and potential impurities in these fully gapped d-wave superconductors. Using analytical T-matrix and complementary numerical tight-binding lattice calculations, we show that magnetic and potential impurities behave fundamentally different in d + is- and d + id′-wave superconductors. In a d + is-wave superconductor, there are no bound states for potential impurities, while a magnetic impurity produces one pair of bound states, with a zero-energy level crossing at a finite scattering strength. On the other hand, a d + id′-wave symmetry always gives rise to two pairs of bound states and only produce a reachable zero-energy level crossing if the normal state has a strong particle-hole asymmetry.

Magnetic impurities on the surface of Rashba spin-orbit-coupled, but otherwise conventional, superconductors provide a promising way to engineer topological superconductors with Majorana bound states as boundary modes. In this work we show that the spin-polarization in the interior of both one-dimensional impurity chains and two-dimensional islands in these systems can be used to determine the superconducting topological phase, as it changes sign exactly at the topological phase transition. Thus, spin polarization offers an alternative method to detect the topological phase in magnetic impurity chains and islands deposited on conventional superconductors, beyond the zero-energy Majorana bound states.

We investigate the impact of topology on the existence of impurity subgap states in a time-reversal-invariant superconductor with an extended s-wave pairing and strong spin-orbit coupling. By simply tuning the chemical potential we access three distinct phases: topologically trivial s-wave, topologically non-trivial s±-wave, and nodal superconducting phase. For a single potential impurity we find subgap impurity bound states in the topological phase, but notably no subgap states in the trivial phase. This is in sharp contrast with the expectation that there would be no subgap state in the presence of potential impurities in s-wave superconductors. These subgap impurity states have always finite energies for any strength of the potential scattering and subsequently, the superconducting gap in the topological s±-wave phase survives but is attenuated in the presence of finite disorder. By creating islands of potential impurities we smoothly connect the single impurity results to topological edge states of impurity island. On the other hand, magnetic impurities lead to the formation of Yu-Shiba-Rusinov states in both the trivial and topological phases, which even reach zero energy at certain scattering strengths. We thus propose that potential impurities can be a very valuable tool to detect time-reversal-invariant topological superconductivity.